Rigidity for W*-McDuff groups

Juan Felipe Arzia Mejia; University of Iowa

A group G is called W*-superrigid whenever it can be fully recovered from its group von Neumann algebra L(G). This phenomenon should be rather rare, as exemplified by the indistinguishability of group factors coming from amenable ICC groups. In between these two extremes of superrigidity, and complete lack thereof, there are many classes of nonamenable groups that display various rigidity phenomena. In our work we introduce the first examples of groups whose lack of superrigidity can be completely characterized. Specifically, we introduce the notion of McDuff W*-superrigid groups, that is groups G such that L(G) = L(H) (for any group H) implies H = G x A for some icc amenable group A. In this talk I will present the two key ingredients for these examples, which are of independent interest: First, the use methods in Geometric Group Theory to construct wreath-like product groups with a 2-cocycle with uniformly bounded support. Second, a new infinite product rigidity result for this class of wreath-like products obtained via the interplay of two deformations. The combination of these two ingredients yields the first examples of McDuff W*-superrigid groups. This is joint work with Ionut Chifan, Denis Osin and Bin Sun.